Distribution of angular misfits in fault-slip data

Abstract

The validity and the accuracy of a stress determination method based on fault-slip data inversion depends on how the misfits are defined and processed in the inversion procedure. To characterize major features of fault-slip data, angular misfits are simulated for both inherited faults and conjugate faults. For one case, a set of 4000 ‘perfect’ fault-slip data was built for the inherited fault type (the general case), with planes randomly distributed on a unit sphere and slip vectors which perfectly fit a given stress tensor. Likewise, a set of 200 ‘perfect’ fault slip data was built for the conjugate fault type, where variations simply occur around two poles. These synthetic fault slip data sets were modified by adding variations of orientations, for both fault planes and slip vectors, according to a Gaussian-type distribution. After re-calculating the stress tensor, we examined the distribution of angular misfits and residuals of five minimization functions (depending on the inversion method adopted), and we evaluated the corresponding variations of the stress tensor. We found that the distribution type of angular misfits is exponential (or half-Gaussian) for both inherited and neoformed fault populations. Iterative or grid search methods are supposed to correspond to a χ 2 test. The application of a χ 2 test to the stress determinations demands that misfits be described as angular misfits which follow a Gaussian law. Thus, to avoid a theoretical contradiction in these iterative or grid search methods, one must adopt a sum of squares misfit criterion as a minimization function. In that the angular misfits follow an exponential law, the function to minimize the sum of the absolute values of misfits can be also meaningful for the stress inversion. The four minimization function misfits considered here are also exponential in type, so that our conclusion apparently contradicts the assumption underlying the iterative linear inversion methods derived by least squares method. The use of least squares criteria in fault-slip data inversion implies that the misfits of the minimization function should follow the Gaussian law. Because the three components of residual vectors follow a Gaussian distribution, we suggest that it is theoretically sound to construct elementary minimization functions using them. The usual form of minimization functions is the coalescence of these three elementary minimization functions. The total inversion method is also consistent with it in that its minimization function is the sum of three elementary minimization functions constructed in three angular differences which follow the Gaussian law. The distribution of the minimization function misfits of the direct inversion method is quite different in type, and can be explained by a β-distribution, which reflects the physical characteristics of the criterion adopted. We evaluated variations in the determined stress tensors as a function of the data dispersion and ‘filtering’ misfit angle. Under the conditions adopted in this paper, the stress solutions depend on the dispersion parameters, and are stable and vary little for smaller data dispersions. The ‘filtering’ misfit angle between 45 ° and 90 ° is appropriate to reconstruct the assumed stress tensor by including as many meaningful fault-slip data as possible.